4.1 The stick-slip vibration characteristics of the system under a constant normal force
The first set of dynamic and static friction coefficients (µs = 0.4, µk = 0.2) is selected to research the stick-slip vibration characteristics of the system under a constant normal force. Figure 7 exhibits the phase diagram and bifurcation diagram under different constant normal forces. The external normal force gradually increases from 10 N to 40 N in steps of 5 N. As the constant normal force increases, stick-slip vibration appears in the system, and the amplitude of the limit cycle of the stick-slip vibration increases sequentially. It is manifest from the Fig. 7(b) that under the condition that the normal force keeps increasing, the balance point of the mass block produces a slight offset, whose value is calculated according to Eq. (6). Under the condition that the normal force keeps increasing, the offset and the amplitude of the mass block increase. The reason is that an increase in the normal force increases the maximum static friction force, inducing the system to generate a greater spring force to get over the static friction. The force ultimately promotes an increase in the adhesion time of the mass block, which stores and releases more energy in a single cycle.
$${x_s}={\mu _k}{F_N}{\text{/}}{k_1}$$
6
The differential equations of the system are solved by MATLAB's ode45 solver. The initial parameter values are different; therefore, the system requires a different number of steps to reach a stable state. In this paper, the duration of each calculation is 100 πs, and the results of the first 300 s are shown in the graph. Only the data after t1 (62.8 s) are selected for the frequency-domain analysis to prevent an influence of the initial value on the analysis of the system motion state. Figure 8 exhibits the time-domain and frequency-domain signals of the stick-slip vibration of the friction system under normal forces of 20 N and 25 N.
Under a constant normal force, the time-velocity curve of the mass shows a constant single-period motion state, and as the normal force increases, the amplitude and period of the system vibration increase. The frequency spectrum of the vibration velocity in t1 ~ 300 s exhibits that the fundamental frequency of the system response reduces with an increase in the normal force. When the normal force is 20 N (25 N), the system produces a multi-order harmonic response with a fundamental frequency of 0.0855 Hz (0.0738 Hz). Table 1lists the fundamental frequencies of the system responses under different constant normal forces. As the normal force increases, the fundamental frequency of the vibration response decreases, and the vibration period of the system increases.
Table 1
The basic frequency of the stick-slip vibration response of the friction system under different constant normal forces
Normal force /FN
|
Response fundamental frequency
|
Normal force / FN
|
Response fundamental frequency
|
10 N
|
0.1243 Hz
|
30 N
|
0.0661 Hz
|
15 N
|
0.1010 Hz
|
35 N
|
0.0583 Hz
|
20 N
|
0.0855 Hz
|
40 N
|
0.0505 Hz
|
25 N
|
0.0738 Hz
|
—
|
—
|
Figure 9 shows the displacement-velocity two-dimensional phase diagram of the mass motion, the three-dimensional phase-space diagram expanded according to the motion cycle (the polar diameter and polar angle of the two-dimensional polar coordinate system respectively represent the displacement and the motion period, and the Z axis represents the speed), and the Poincaré cross-section based on the projection of the three-dimensional phase-space trajectory on a plane with a polar angle of π and parallel to the Z-axis (O in the figure represents the pole of the polar coordinates, and θ represents the polar angle). The Poincaré cross-section diagram is used to distinguish the periodic motion, quasi-periodic motion, chaotic motion, and other motion behaviors of the system according to the dynamic differential equations of nonlinear systems. Under the constant normal force, the motion state of the system shows stable single-period stick-slip vibration. The two-dimensional phase diagram indicates a single-period limit cycle, which is expanded in the three-dimensional phase-space with only one motion trajectory; thus, only one point is shown in the Poincaré cross-section in Fig. 9(c). Therefore, under the constant forces of 20 N and 25 N, the masses exhibit single-period stick-slip motion.
4.2 The stick-slip vibration characteristics of the system under an alternating normal force
Figure 10 shows the bifurcation diagram of the displacement of the mass with the angular frequency of excitation at a median alternating normal force of F0 = 25 N and an amplitude of Fω=5 N. As the exciting angular frequency increases, the mass exhibits multiple motion states, such as chaos, single-period vibration, and multi-period vibration. The bifurcation diagram is divided into seven regions. In region 'I', the movement state of the system is disordered; in regions 'II', 'IV', and 'VI', the system is in a single-period motion state; in regions 'III', 'V', and 'VII', the system exhibits multiple motion states, including single-period, multi-period, quasi-period, and chaos states. Non-classical bifurcation phenomena are observed near critical points, such as sudden boundary changes and jumps. For example, when the excitation angular frequency is 1.35 rad/s (the critical point of regions 'IV' and 'V'), the single-period motion state of the mass block suddenly changes.
Further, we discuss the stick-slip vibration characteristics of the friction system for different angular frequencies. An excitation angular frequency of ω = 0.27 rad/s is selected in the chaotic stage, and ω = 0.5 rad/s is selected in the single-period stage, and ω = 0.69 rad/s and ω = 1.65 rad/s are chosen in the vibration stage.
Figure 11 displays the phase diagram, phase-space diagram, and Poincaré cross-section diagram of the stick-slip vibration of the friction system at different angular frequencies. At ω = 0.27 rad/s, multiple limit cycles are observed in the phase diagram. As the calculation time increases, the number of limit cycles increases, and the phase-space trajectory becomes more chaotic. Multiple discrete points are observed, and the system is in chaotic stick-slip motion at this time. At ω = 0.5 rad/s, the phase diagram exhibits a single-period limit cycle, the phase-space diagram has only one trajectory, and only one point is visible in the Poincaré cross-section diagram. Under this circumstances, the system is in a single-period stick-slip motion state. At ω = 0.69 rad/s, the system is in a two-period vibration state. The phase diagram and phase-space diagram show an additional trajectory, and the Poincaré cross-section diagram exhibits two discrete points. At ω = 1.65 rad/s, the Poincaré cross-section depicts a straight line, and the system is in a quasi-periodic stick-slip state.
Figure 12 shows the external excitation signal, vibration velocity signal, and fast Fourier transform (FFT) analysis results of the system for four excitation angular frequencies. The value of t1 is the same in Fig. 12 and Fig. 8. In Fig. 12(c), the left Y-axis depicts the FFT results of the speed signal, and the right Y-axis shows the stress vibration frequency (dashed line). The friction system produces multiple main response frequencies at an excitation angular frequency of 0.27 rad/s and an excitation frequency of 0.0430 Hz. The highest response frequency of the system is 0.0725 Hz, and the vibration behavior of the mass block is complex. At 0.5 rad/s, the friction system produces a multi-order harmonic response with a fundamental frequency of 0.0796 Hz. At this time, the excitation frequency is consistent with the response frequency of the system with the highest energy, and the system is in a single-cycle stick-slip state (Fig. 10). At 0.69 rad/s, the excitation frequency is 1.5 times of the response frequency of the system with the highest energy. At this time, the system is in a two-period stick-slip state. At 1.65 rad/s, the excitation frequency is in keeping with the third-order vibration frequency component of the system’s response fundamental frequency, and the system is in a quasi-periodic stick-slip motion state.
Further, we select an excitation angular frequency in each region in Fig. 10 and calculate the response frequency of the system with the highest energy. The frequency data is exhibited in Table 2. The excitation angular frequency of 0.67 rad/s in area 'III is greater than the highest response frequency but less than the second-order response frequency of the system. The value is three-half of the main frequency of the highest response of the system. At this time, area 'III' is the transition area between the single-period stick-slip motion areas 'II' and 'IV'. In area 'IV', the excitation angular frequency is 1 rad/s, which is consistent with the second-order response frequency of the system. The system is in a single-period stick-slip state. In region 'V', the excitation angular frequency is 1.7 rad/s, which is consistent with the third-order response frequency of the system. However, Fig. 10exhibits that the system is in a multi-period stick-slip motion state at this time. In area 'VI', the excitation angular frequency is 2.25 rad/s, which is consistent with the fourth-order response frequency of the system; the system is still in a single-period stick-slip motion state. In area 'VII', the excitation angular frequency is 3.15 rad/s, which is equal to the fifth-order response frequency of the system. As the excitation frequency increases, the system goes through different vibration states, including single cycle, multi-period, quasi-period, and chaotic vibration states.
Table 2
Response frequency of the system for different excitation angular frequencies
|
Area
|
Excitation angular frequency ω(rad/s)
|
Excitation angular frequency(Hz)
|
The main frequency with the highest energy(Hz)
|
Median force: 25 N
Excitation amplitude: 5 N
|
I
|
0.20
|
0.0318
|
0.0732
|
II
|
0.52
|
0.0828
|
0.0828
|
III
|
0.67
|
0.1066
|
0.0720
|
IV
|
1.00
|
0.1592
|
0.0796
|
V
|
1.70
|
0.2706
|
0.0902
|
VI
|
2.25
|
0.3581
|
0.0901
|
VII
|
3.15
|
0.5013
|
0.1003
|
These results show that the condition of the system is chaotic vibration when the excitation magnitude is constant, the excitation frequency does not cause a harmonic response of the system, and the excitation frequency is not a multiple of the main frequency of the system with the highest energy response. When the excitation frequency causes a harmonic response of the system, the system is in a single-cycle motion state if the excitation frequency is in keeping with the main frequency of the system with the highest energy response or an even-order multiple (second-order, fourth-order) of the main frequency. The system can have various vibration states if the frequency is the same as the dominant frequency of the highest odd-order (third-order) of the system, or the excitation frequency is greater than the dominant frequency of the higher-order (fifth-order) response of the system.
Figure 13 shows the bifurcation diagram of the mass displacement with the excitation angular frequency for an excitation amplitude value of Fω=10 N. Increasing the excitation amplitude increases the displacement extremum of the bifurcation diagram of the system, which agrees with the results in Fig. 7. The bifurcation diagram is divided into three regions, and the main frequency of the system response for different excitation angular frequencies is analyzed. The results are listed in Table 3. In area 'I', the excitation angular frequency is 0.15 rad/s. This excitation frequency does not cause a harmonic response of the system and is less than the main frequency response of the system with the highest energy. The movement state of the system is disordered. In area 'II', the excitation frequency is in keeping with the dominant frequency of the system’s highest energy response, and the system is in a single-cycle stick-slip state. Two excitation angular frequencies (1.35 rad/s and 4.15 rad/s) are selected in area 'III'. At 1.35 rad/s, the excitation frequency is equal to the third-order response frequency of the system, and at 4.15 rad/s, the excitation frequency is in keeping with the seventh-order response frequency of the system. In area 'III', the motion state of the system changes from a single cycle to two cycles and ultimately enters a chaotic motion state as the excitation angular frequency increases. These results indicate that an increase in the excitation amplitude of the alternating normal force increases the area where the system is in multiple vibration states; the system is more likely to be in a state of chaotic motion.
Table 3
Response frequency of the system for different excitation angular frequencies
|
Area
|
Excitation angular frequency ω(rad/s)
|
Excitation angular frequency(Hz)
|
The main frequency with the highest energy(Hz)
|
Median force: 25 N
Excitation amplitude: 10 N
|
I
|
0.15
|
0.0239
|
0.0743
|
II
|
0.55
|
0.0875
|
0.0875
|
III
|
1.35
|
0.2149
|
0.0716
|
4.15
|
0.6605
|
0.1247
|
The influence of the excitation amplitude value on the system motion state is analyzed for an excitation angular frequency of 0.5 rad/s. The bifurcation diagram of the mass displacement response is shown in Fig. 14. The bifurcation graph is divided into two regions. In region 'I', the system is in a multi-period stick-slip motion state, and the excitation frequency does not cause a higher energy response frequency of the system. In region 'II', the system is in a single-period stick-slip motion status, and the excitation frequency is in keeping with the response frequency of the system with the highest energy (see Table 4for the details).
Table 4
Response frequency of the system for different excitation amplitudes
|
Area
|
Excitation amplitude Fω(N)
|
Excitation frequency
ω(Hz)
|
The main frequency with the highest energy(Hz)
|
Median force: 25 N
Excitation angular frequency: 0.5 rad/s
|
I
|
0.8
|
0.0796
|
0.0756
|
II
|
5.0
|
0.0796
|
0.0796
|
10.0
|
0.0796
|
0.0796
|
Figure 15 exhibits the bifurcation diagram of the displacement response of the mass with the excitation amplitude at an excitation angular frequency of 1.65 rad/s. Similarly, the bifurcation graph is divided into three regions. The vibration form of the friction system is more complex for the excitation angular frequency of 1.65 rad/s than of 0.5 rad/s. In area 'I', the system is in a state of multi-period stick-slip motion. At this time, the external excitation frequency does not cause a multi-order harmonic response of the system and a higher energy response of the main frequency. In area 'II', the system is in a single-period stick-slip state, and the excitation frequency is equal to the fourth-order response frequency of the system. The displacement response of the mass block undergoes abrupt changes and increases at the critical points of regions 'II' and 'III'. In area 'III', the system is in a variety of vibration states prior to a normal force of 13.1 N. Subsequently, the system begins to bifurcate into two motion states, which can be approximated as a combination of two multi-period motions.
Different excitation amplitude values in area 'III' are chosen to calculate the response frequency of the system, as listed in Table 5. A state of motion exists where the excitation frequency stimulates the harmonic response of the system and is in keeping with the third-order response frequency. A harmonic response whose excitation frequency does not affect the system is also observed, resulting in a state of motion in keeping with the highest response frequency of the system. In addition, the friction system also has a two-period-like motion state in which two multi-period motions are combined, and the excitation frequency of the system is in keeping with the second-order response frequency of the system. At the same excitation amplitude, increasing the excitation frequency complicates the system's vibration form, and it becomes more difficult to transition to a single-period stick-slip state.
Table 5
Response frequency of the system for different excitation amplitudes
|
Area
|
Excitation amplitude Fω(N)
|
Excitation frequency
ω(Hz)
|
The main frequency with the highest energy(Hz)
|
Median force: 25 N
Excitation angular frequency: 1.65 rad/s
|
I
|
1.0
|
0.2626
|
0.1472
|
II
|
2.8
|
0.2626
|
0.1313
|
III
|
4.2
|
0.2626
|
0.0875
|
5.1
|
0.2626
|
0.0875
|
7.5
|
0.2626
|
0.0915
|
8.5
|
0.2626
|
0.0875
|
10.0
|
0.2626
|
0.2626
|
11.5
|
0.2626
|
0.2626
|
14.0
|
0.2626
|
0.1313
|